The Modified Weak Galerkin Finite Element Method For Second-order Elliptic Equations And Linear Elasticity Problem In Mixed Form

A modified weak finite element method(MWG)which is improved by the weak finite element method is introduced in this paper.The weak finite element method is often used to solve the related problems of partial differential equations.The main idea is to replace the traditional differential operator with a specially defined weak differential operator.The discontinuous piecewise polynomials are chosen as the bases in the element interior and boundary accordingly,so the continuity between the two units isn't necessary,however,that increases the unknown numbers of the whole discrete system.Comparing with the weak finite element method,the advantage of the modified weak finite element method is that boundary function on each element is defined by the corresponding interior function,so it has less freedom and improves the computational efficiency.In this paper,we use the MWG method to solve the second order elliptical equations and the mixed linear elastic equations.By defining the special weak divergence operator to replace the divergence operator in the traditional sense,we introduce the stabilizer and give the corresponding variation form in the weak function space.Then,error equations and the optimal order error estimations in H~1norm and L~2norm are obtained,respectively.Finally,the correctness of the error estimation of the linear elastic equation is verified by numerical experiments.